
The geometrical interpretation of spinors.
In order to understand the geometrical interpretation of the newly constructed spinfoam amplitude, it is enough to unravel the geometrical meaning of the holomorphic inner product of spinors, \(\textit {i.e.}\)  the angle couplings. It would be interesting to further understand the geometrical interpretation of the spinor itself. Bearing in mind the relation of the spinors and flux vectors, most importantly the equivalence of the spinor norm and the norm of the flux vector it describes (see (REF )), we expect that the spinors encode the length/scale information, which is washed out by constructing the holomorphic inner product.

The Wheeler de-Witt equations of the SGF.
Recall that in the original Ponzano-Regge state-sum model, the invariance of the amplitude under Pachner moves is guaranteed by the Biedenharn-Elliott identity of the \(\lbrace 6j\rbrace \) -symbols, which is closely related to the recursion relations of the \(\lbrace 6j\rbrace \) -symbols. These recursion relations can be viewed as generated by the Wheeler de-Witt equations of the spin network states. It would be interesting to explore whether the invariance of the newly constructed amplitude formalism under Pachner moves can be directly proven by (possibly the deformation of) the Wheeler-de Witt equations of the SGF. If possible, it would provide a new way that is similar to the original state-sum model to see the topological invariance of the new spinfoam formalism, directly from the scale-invariant factorization of the amplitude, \(\textit {i.e.}\)  the second line of (REF ). As an intermediate step, it may be useful to rewrite the Wheeler de-Witt equation in terms of the link couplings so that variables are more decoupled. Similar to the differential equations found in [1]}, one is able to find four such equations, each associated to one triangle on the boundary of the tetrahedron and only three are independent. An example is (referring to the notation in fig.REF )
\(\begin{split}&3 \left[Y_1\left(\frac{\partial {\mathcal {S}}}{\partial Y_2 }\frac{\partial {\mathcal {S}}}{\partial Y_6 }- \frac{\partial {\mathcal {S}}}{\partial Y_3 }\frac{\partial {\mathcal {S}}}{\partial Y_5 }\right)+ Y_2\left(\frac{\partial {\mathcal {S}}}{\partial Y_3 }\frac{\partial {\mathcal {S}}}{\partial Y_4 }- \frac{\partial {\mathcal {S}}}{\partial Y_1 }\frac{\partial {\mathcal {S}}}{\partial Y_6 }\right)+ Y_3\left(\frac{\partial {\mathcal {S}}}{\partial Y_1 }\frac{\partial {\mathcal {S}}}{\partial Y_5 }- \frac{\partial {\mathcal {S}}}{\partial Y_2 }\frac{\partial {\mathcal {S}}}{\partial Y_4 }\right)\right]\\-&2 {\mathcal {S}}\left[Y_1\left(\frac{\partial ^2 {\mathcal {S}}}{\partial Y_2 \partial Y_6 }- \frac{\partial ^2 {\mathcal {S}}}{\partial Y_3 \partial Y_5 }\right)+ Y_2\left(\frac{\partial ^2 {\mathcal {S}}}{\partial Y_3 \partial Y_4 }- \frac{\partial ^2 {\mathcal {S}}}{\partial Y_1 \partial Y_6 }\right)+ Y_3\left(\frac{\partial ^2 {\mathcal {S}}}{\partial Y_1 \partial Y_5 }- \frac{\partial ^2 {\mathcal {S}}}{\partial Y_2 \partial Y_4 }\right)\right]=0\,,\end{split}\) 
where \({\mathcal {S}}\)  is the SGF and \(Y_i\)  the link coupling of link \(l_i\) .
On the other hand, as the SGF not only describes a flat tetrahedron but a scale-invariant one, it should be possible to determine a different differential equation that generates the scale-invariance of the SGF. The equation to be found describes the dilatation behaviour of the SGF, thus playing a dynamic role and can also be viewed as the Wheeler de-Witt equation. The flow of the differential equation should generate the symmetry of the link couplings (REF ) in terms of spinors, namely the \(\mathrm {SL}(2,{\mathbb {C}})\)  symmetry and the “anti-rescaling” symmetry (REF ).

Spinfoam model for conformal quantum gravity. 
The building blocks of the new spinfoam are scale-invariant, thus almost but not yet conformal invariant. It would be interesting to push forward to construct a spinfoam with conformal blocks, whose boundary states are conformal invariant. Unfortunately, we have no clue whether it can be constructed with the spinor representation or what structures of the vertex amplitude we should expect. It may be more reliable to start the journey from the discretization of conformal gravity action and construct the partition function applying the action principle [2]}. Such a spinfoam model, if exists, would be a more suitable framework to study the CFT/gravity duality at the discrete level.

Group field theory for 3D quantum gravity with spinors. 
It has been well-known that the Ponzano-Regge models can be generated by a group field theory (GFT) with \(\mathrm {SU}(2)\)  group and the amplitude is interpreted as some Feynman graph evaluation [3]}. GFT reformulation with spinors of the Ponzano-Regge model can capture the full \(\mathrm {SU}(2)\)  structure rather than \(\mathrm {SO}(3)\)  when using the flux and holonomy variables [4]}, [5]}. An attempt was made in [6]}
where the amplitude is not yet separated into local blocks. In order to build a similar connection of the vertex amplitude with the interaction of the GFT and of the edge amplitude with the propagator of the GFT, a better starting point is a spinfoam formulation with local amplitudes, thus the result of Proposition REF . We expect that this GFT, if found, also has a scale-invariant nature, thus it may be better suited to study the renormalization properties in quantum gravity.

Generalization to a 4D spinfoam. 
The spinfoam model built with spinors has been applied to 4D BF theory [7]}, [8]}, [9]}. However there remains spin dependence or \(\mathrm {SU}(2)\)  holonomy dependence of the local amplitude in these existing models. It would be interesting to discover a 4D spinfoam with local amplitudes given only by the spinor variables and explore what these local amplitudes represent geometrically. A natural guess is that the vertex amplitude could be constructed by the generating function of the \(15j\) -symbols [10]}, which can also be written into a closed form with a loop structure as the SGF.

Including the cosmological constant. 
The spinor variables can be deformed to describe the loop gravity with a negative cosmological constant [11]}, [12]}, which recovers the loop gravity framework described by deformed holonomy and flux variables [13]} (see also [14]} for its quantization). It would be interesting to see if the deformed spinors can be used to construct the spinfoam that recovers the Turaev-Viro model [15]}, whose building blocks are the \(q\) -deformed \(\lbrace 6j\rbrace \) -symbols. One of the first things to do is to construct the “\(q\) -deformed scaleless spin network statl” on the hyperbolic geometry. We expect that the spinorial framework of LQG and the spinfoam model can be generalized to a \(q\) -deformed version with the use of quantum groups and describe the quantum gravity with a non-vanishing cosmological constant.

